α β 为锐角sin^4α/cos^2β+cos^4α/sin^2β=1,证α+ β=π/2
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α β 为锐角sin^4α/cos^2β+cos^4α/sin^2β=1,证α+ β=π/2
∵(sinα)^4/(cosβ)^2+(cosα)^4/(sinβ)^2=1,
∴(sinα)^4(sinβ)^2+(cosα)^4(cosβ)^2=(sinβ)^2(cosβ)^2,
∴[1-(cosα)^2]^2(sinβ)^2+(cosα)^4(cosβ)^2=(sinβ)^2(cosβ)^2,
∴(sinβ)^2-2(cosα)^2(sinβ)^2+(cosα)^4(sinβ)^2+(cosα)^4(cosβ)^2
=(sinβ)^2(cosβ)^2,
∴(sinβ)^2-2(cosα)^2(sinβ)^2+(cosα)^4[(sinβ)^2+(cosβ)^2]
=(sinβ)^2(cosβ)^2,
∴(sinβ)^2[1-(cosβ)^2]-2(cosα)^2(sinβ)^2+(cosα)^4=0,
∴(sinβ)^4-2(cosα)^2(sinβ)^2+(cosα)^4=0,
∴[(sinβ)^2-(cosα)^2]^2=0,
∴(sinβ)^2=(cosα)^2,
∵α、β都是锐角,∴sinβ>0、cosα>0,∴sinβ=cosα,∴α+β=π/2.
∴(sinα)^4(sinβ)^2+(cosα)^4(cosβ)^2=(sinβ)^2(cosβ)^2,
∴[1-(cosα)^2]^2(sinβ)^2+(cosα)^4(cosβ)^2=(sinβ)^2(cosβ)^2,
∴(sinβ)^2-2(cosα)^2(sinβ)^2+(cosα)^4(sinβ)^2+(cosα)^4(cosβ)^2
=(sinβ)^2(cosβ)^2,
∴(sinβ)^2-2(cosα)^2(sinβ)^2+(cosα)^4[(sinβ)^2+(cosβ)^2]
=(sinβ)^2(cosβ)^2,
∴(sinβ)^2[1-(cosβ)^2]-2(cosα)^2(sinβ)^2+(cosα)^4=0,
∴(sinβ)^4-2(cosα)^2(sinβ)^2+(cosα)^4=0,
∴[(sinβ)^2-(cosα)^2]^2=0,
∴(sinβ)^2=(cosα)^2,
∵α、β都是锐角,∴sinβ>0、cosα>0,∴sinβ=cosα,∴α+β=π/2.
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